Answer:
We conclude that he coach can choose 252 lineups.
Step-by-step explanation:
We have three pure centers, four pure forwards, four pure guards and one swing- man who can play either guard or forward.
First case: swing- man play guard.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 1.
![C_1^3\cdot C_2^4\cdot C_1^4=3\cdot \frac{4!}{2!(4-2)!}\cdot 4=12\cdot 6=72](https://tex.z-dn.net/?f=C_1%5E3%5Ccdot%20C_2%5E4%5Ccdot%20C_1%5E4%3D3%5Ccdot%20%5Cfrac%7B4%21%7D%7B2%21%284-2%29%21%7D%5Ccdot%204%3D12%5Ccdot%206%3D72)
Second case: swing- man play forward.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 1, and from 4 from the guards to choose 2.
![C_1^3\cdot C_1^4\cdot C_2^4=3\cdot 4\cdot \frac{4!}{2!(4-2)!}=12\cdot 6=72](https://tex.z-dn.net/?f=C_1%5E3%5Ccdot%20C_1%5E4%5Ccdot%20C_2%5E4%3D3%5Ccdot%204%5Ccdot%20%5Cfrac%7B4%21%7D%7B2%21%284-2%29%21%7D%3D12%5Ccdot%206%3D72)
Third case: swing- man not play.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 2.
![C_1^3\cdot C_2^4\cdot C_2^4=3\cdot \frac{4!}{2!(4-2)!}\cdot \frac{4!}{2!(4-2)!}=3\cdot 6 \cdot 6=108](https://tex.z-dn.net/?f=C_1%5E3%5Ccdot%20C_2%5E4%5Ccdot%20C_2%5E4%3D3%5Ccdot%20%5Cfrac%7B4%21%7D%7B2%21%284-2%29%21%7D%5Ccdot%20%5Cfrac%7B4%21%7D%7B2%21%284-2%29%21%7D%3D3%5Ccdot%206%20%5Ccdot%206%3D108)
Now, we get 72+72+108=252.
We conclude that he coach can choose 252 lineups.